Which one of the following metals will not react with cold water but will react with steam?

You must keep in mind that some of the listed metals react with water to produce the corresponding oxides and hydroxides in order to correctly answer this question. But when combined with steam, the transition metals in the list of choices do react, producing the respective oxides and hydrogen gas.

Both hot and cold water have no effect on the reaction with zinc. It creates a coating of zinc hydroxide $\left[\mathrm{Zn}{\left(\mathrm{OH}\right)}_2\right]$ that acts as a protective barrier. However, it strongly interacts with steam and produces hydrogen gas and zinc oxide. It is possible to write the reaction's chemical equation as-
$\mathrm{Zn}+{\mathrm{H}}_2{\mathrm{O}}_{\left(\mathrm{steam}\right)}\to \mathrm{ZnO}+{\mathrm{H}}_{2\left(\mathrm{g}\right)}$
Cold or hot water has no effect on iron either. However, it also slowly interacts with steam, producing hydrogen gas and iron oxide. The reaction can be expressed as-
$\mathrm{Fe}+{\mathrm{H}}_2{\mathrm{O}}_{\left(\mathrm{g}\right)}\to \mathrm{FeO}+{\mathrm{H}}_{2\left(\mathrm{g}\right)}$
Cold water has a very sluggish (slow) reaction with Calcium. It doesn't directly react to steam. With water, it produces hydrogen gas and calcium hydroxide. The response is-
$\mathrm{Ca}+2{\mathrm{H}}_2{\mathrm{O}}_{\left(\mathrm{g}\right)}\to \mathrm{Ca}{\left(\mathrm{OH}\right)}_2+{\mathrm{H}}_{2\left(\mathrm{g}\right)}$
 

Hence, the correct answer is option A.